Irreducible Polynomials:

**joanne_q** · March 3rd 2008, 09:37 AM

(1):
Find all irreducible polynomials of the form $x^2 + ax +b$ , where a,b belong to the field $\mathbb{F}_3$ with 3 elements.
Show explicitly that $\mathbb{F}_3(x)/(x^2 + x + 2)$ is a field by computing its multiplicative monoid.
Identify [ $\mathbb{F}_3(x)/(x^2 + x + 2)$ ]* as an abstract group.

any suggestions please?

**ThePerfectHacker** · March 3rd 2008, 10:07 AM

Originally Posted by joanne_q

(1):
Find all irreducible polynomials of the form $x^2 + ax +b$ , where a,b belong to the field $\mathbb{F}_3$ with 3 elements.

Just list all of them $x^2 + x+1,x^2+x+2,...$ there are only $9$ or them. Now, just check which ones have zeros (there are only three zeros to check). And this tells you which are irreducible and which are not.

Show explicitly that $\mathbb{F}_3(x)/(x^2 + x + 2)$ is a field by computing its multiplicative monoid.
Identify [ $\mathbb{F}_3(x)/(x^2 + x + 2)$ ]* as an abstract group.

Any element in $\mathbb{F}_3[x]/(x^2+x+2)$ has form $a+bx+(x^2+x+2)$ . Now show that these elements form a field.

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